Friday, September 28, 2012

The F.E.D. Dialectic of the Standard Arithmetics, Translated into a Dialogue

Full Title:   The Opening Stages of the The F.E.D. Dialectical 'Meta-Model' of the 'Meta-Systematic Dialectic' of the Systems of the Standard Arithmetics, Translated into a "Q&A" 'Dialogue-ic' Format.


Dear Readers,

Below I have extracted, for your reading pleasure and intellectual enjoyment, the section from pages II-11 through II-14 of F.E.D. Vignette #4, The Goedelian Dialectic of the Standard Arithmetics.


URLs --

http://www.dialectics.org/dialectics/Welcome.html


http://www.dialectics.org/dialectics/Vignettes.html


http://www.dialectics.org/dialectics/Vignettes_files/v.1,Part_II_of_II,Miguel_Detonacciones,F.E.D._Vignette_4,The_Goedelian_Dialectic_of_the_Standard_Arithmetics,posted_20SEP2012.pdf





The section quoted below is exemplary as to how other dialectical-mathematical 'meta-models', formulated using the NQ_ dialectical algebra, may be translated into the format of a simple Question/Answer dialogue, based upon concepts sourced from the tradition of "Systematic Dialectics" methods of presentation.



"In resonance with the prologue to the quote from Nicholas Rescher extracted in section B.gamma. of Part I.: “Already the Socrates of Plato’s Theaetetus conceived of inquiring thought as a discussion or dialogue that one carries on with oneself.” [ibid., p. 46], the ‘Dyadic Seldon Function ‘meta-model’ central to this essay can be most directly interpreted -- especially taking into account what we have had to say above regarding ‘‘‘personification’’’ and '''impersonation'''-- as ‘meta-modeling’ a generic such self-dialogue’, as a readily reusable ‘recipe for thought’ for a systematic and well-ordered refresher for one’s self, and for re-presentation to others, of the contemporary “Standard Arithmetics”.

However, in Part I., version 2, we translated our eight-category Dyadic Seldon Function model for the systematic dialectic of the Value-Form content in Marx’s Capital into the narrative format of an ‘exo-allo-dialogue’ between two generic, fictive interlocutors, rather than into an ‘endo-auto-dialogue’ within the mind of a single such generic, fictional “character”.

We can do likewise with our Dyadic Seldon Function ‘meta-model’ presentation for the ‘meta-system’ of the “Standard Arithmetics”, translating it into a somewhat Socratic, ‘dialogue-ic’ “Q&A”, in a way which perhaps more compellingly reveals what is implicitly involved in the Seldon-Function categorial progression algorithm, as applied to this subject-matter.


A sample of such a translation is exhibited below.


In reviewing the dialogue below, it is important to keep in mind that the ‘‘‘analysis of categories’’’, referenced repeatedly therein, has a special meaning in this context.

The ‘‘‘analysis of a category’’’ means, herein, the process of elaboration of the content of that category, in the sense of the ‘explicitization’ of at least some of its content which would otherwise remain implicit, i.e., if only that category’s name were given: the explicit evocation of some, at least, of the sub-categories of that category, and, perhaps, of some of their sub-categories as well, i.e., the elaboration of at least some of the category’s ‘sub-sub-categories’, or ‘sub^2-categories’, as well.

‘‘Analysis’’ in this, dialectical, sense, is the process of human cognition by which we “ascend” [Marx, reversing dialectical predecessors Plato, et al.] to greater detail, or to greater “determinateness” and ‘thought-concreteness’, from greater abstraction and abstract simplicity.

It is the oppositely-directed cognitive movement/movement-of-cognition to that of ‘‘‘Synthesis’’’, whereby we “descend” from multiple, more detailed, more specific ‘sub^n-categories’, to a single, more general, more generic, more “abstract” and simplified ‘sub^(n-1)-category’, one that embraces, and ‘[re-]implicitizes’ into itself, all of the ‘sub^n-categories’ immediately ‘“above”’ it, as well as all of those ‘sub^(n+1)+-categories’ ‘“above”’ them [if any].



Q1:    ¿What is the simplest category that grasps the totality of our -- and, in our view, of most people’s -- ‘untheorized’, ‘unsystematic’, “chaotic” [Marx] experience, and knowledge, of the modern / contemporary Standard Arithmetics?

A1:   The category of the “Natural system of arithmetic, the system which is about the  

N = { I, II, III, ... }

set of numbers, and whose analysis is as follows: ... . It is the system that ‘explicitizes’ "Standard Numbers" as “counts” -- as cardinal numbers.


Q2:   ¿Does the analysis of this arithmetic-system category, N#, exhaustively systematize our previously fragmentary experience and “chaotic” knowledge of the totality of Standard Arithmetic, accounting entirely for/explaining all of the ‘ideo-phenomena’ of number that we encounter today?   ¿Or, are there other categories needed to entirely comprehend that experience and knowledge, to fully classify/systematize and explain those ‘ideo-phenomena’ of number; categories whose content is not explicitly covered by the analysis of the “Natural Arithmetic” system-category; categories whose analysis could therefore add to our systematic grasp of the totality of the systems of modern/contemporary “Standard Arithmetic”?  We offer into evidence here the N-algebraic equation

n
+ x1 =
n
, for all n in N,

not satisfiable by any number in N: x1 is not an element of N.

And this equation is paradoxical from the point of view of the N# definition of ‘Standard Number’, of number “Natural-ness, for which addition cannot result in no increase in numeric magnitude.

A2:   The category of the arithmetical sub-system of the ‘‘‘zeros’’’, or of the ‘aught numbers --

a
=
{ I-I = 0, II-II = 0, III-III = 0, ... } --

the sub-system category denoted by a# -- is needed to satisfy the kind of equation that you cited.  Your equation transforms, algebraically, to --

x1
= n - n
,

-- and so invokes the set of results of self-subtraction for all “Naturals”, n.  The analysis of the category of the aughts ‘explicitizes’ an expansion of the meaning/definition of ‘Standard Number’, over and above that explicit in the ‘N-numbers’ concept:   . . ..

The upshot of this analysis is that not just “counts”, but also no[n]-counts’ -- certain kinds of not-counts’ -- are also among the ‘Standard Numbers’.


Q3:   ¿Do the [sub-]system categories of the “Naturals and the ‘aughts, of “counts” as ‘Standard Numbers’ versus of no[n]-counts’ as ‘Standard Numbers’, simply represent two irreconcilable, absolutely disparate and separated and, in some ways, diametrically opposite categories? ¿Or, do we find, in our experience of the human number meme, that these two categories interrelate, and even combine? ¿That is, does the no[n]-counts’ category, the counter-example to, and the breakdown of, the “counts” concept/definition of ‘Standard Number’, unite with the category that it counters, to form a new whole system-of-arithmetic category? ¿Can our explanation/systematization/classification of the ‘meta-system’ of the “Standard Arithmetics” be adequate as either the analysis of the category N# by itself, or the analysis of category a# by itself, as two, distinct, disjunct alternatives?

A3:   “Yes”, the “Naturals and the ‘aughts are two different, separate, disparate, even qualitatively opposite categories of kinds of number, of number ‘ideo-ontology’.  But “No”, they are not absolutely separate, or absolutely separable, in our experience of humanity’s numbers meme.  And our theory/presentation of the “Standard Arithmetics” ‘meta-system’ is inadequate to that experience if it takes a# and N# as only dirempt.  Indeed, in our experience/knowledge of modern/contemporary Arithmetics, they both are ingredient in, and inseparable as, the vastly superior numeration system -- vastly superior to the Roman Numerals and to the other ancient, additive/'juxtapositional' numeration-systems -- that is the Indo-Arabic numeration system, the "place-value" numeration system made possible by the advent of 0 as "place-holder", and as full number in its own right, in which separate a# and N# give way to their ‘“complex unity”’, to the W# system which the former form by their ‘“combination”’.  The ‘‘‘combination’’’ of the “Naturals and the ‘aughts constitutes the “Wholes -- the “Whole numbers”, such that

W
= { 0...0, 0...01, 0...02, 0...03, ... },

and in which the equation

w
+ x1 = w
, for all w in W,

which is a ‘‘‘well-formed equation’’’ in W#, can “now” be satisfied, namely, by

x1
= 0
, x1 is an element of W.

The analysis of the W# category yields a concept/definition of ‘Standard Number’ that encompasses and reconciles the “counts” & no[n]-counts’ definitions of number, in a single concept/definition of number ‘‘‘Whole-ness’’’.


Q4:    ¿Does the analysis of this new, composite, successor system-of-Standard-Arithmetic category, W#, exhaust our experience and knowledge of the totality of contemporary/modern Standard Arithmetic(s)? ¿Does that analysis account entirely for, i.e., does it explain, that experience and knowledge?  ¿Or are there other categories that are required to cover, and to systematize, that experience and knowledge; required to classify, and to explain, all of it -- or, at least, more of it: including even [at least some of] the most exotic ‘ideo-phenomena’ of modern/contemporary Standard Arithmetic(s) that we have encountered; other categories whose content is not explicitly elaborated by the analysis of the “Whole-numbers arithmetic-system category; other categories whose analysis/‘content-explicitization’ would therefore add to our comprehension of the totality of modern/contemporary Standard Arithmetic(s)?   I call your attention, specifically, to the algebraic equation

w
+
x2 = 0,

0 & w in W, which is a ‘‘‘well-formed equation’’’ in W# [i.e., ‘w’, ‘+’,‘x’, ‘2’ [including as a subscript], ‘=’, & 0’ are all parts of the “official vocabulary” of W#].  Despite this equation’s being an equation of  W# -- a ‘‘‘well-formed’’’, “legal” equation in W# [though not so in N#, because 0 is not in N] -- this equation is not satisfiable by any number in W:   x2 is not in W.  This equation defines an ‘ideo-phenomenon’ of ‘decreasive addition’, one which is paradoxical from the point of view of the new, W#, definition of ‘Standard Number’, because the possibility of ‘decreasive addition’ is not instantiated within W#.  The nature of number as implied and specified in W# is contradicted by the “number” whose existence is implied by this equation, under the name of its “unknown”, x2.  The behavior ascribed, by this equation as a whole, to the number, x2, is of a kind-of-number(s) which, when added to any W-number, “nullifies” it, decreasing it all the way back to 0.

A4:   The analysis of the W# composite category as a whole, and of each of its constituent sub-categories -- N#, a#, & #qaN -- which, taken together, constitute/connote W#, analyzed again, individually, but now in the context of the analysis of W# as a whole, does further advance our detailed comprehension of our experience and knowledge of the modern/contemporary Standard Arithmetic(s), #. But, all of that analysis still leaves out[side of itself] fundamental aspects of our experience and knowledge of the ‘ideo-phenomena’ of this ‘human-phenomic’ totality that we ‘‘‘name’’’ #.

In particular, the category of the axiomatic system-component for the “minus numbers”, m#, is needed to solve the equation

w
+
x2 = 0, in which x2 functions as an “additive inverse” of w.

The analysis of this “minus numbers” category is needed to advance our comprehension of the modern/contemporary Standard Arithmetic(s) beyond the ken of the W# category. The analysis of the axiomatic sub-system category of the “minus numbers”, or of the “negative[ly]-signed” numbers, such that


m = { ±0-I = -1, ±0-II= -2, ... },


'explicitizes' the following expansion of the meaning -- of the definition -- of the ‘Standard Numbers’: . . ..

The upshot of this expansion is that numbers, and that counting, and that ‘‘‘counting numbers’’’, can have direction-- can have one of two ‘‘‘co-linear directionalities’’’, in terms of the number-“line” convention of the ‘‘‘analytic-geometric’’’ visualization of the ‘‘‘number-spaces’’’ of these number-systems: an explicit right-hand directionality [labeled by the ‘+’ sign], or an explicit left-hand directionality [labeled by the ‘-‘ sign], or, in just a "single" case -- the case of ±0 -- directional ‘‘‘neutrality’’’/intra-bi-ality’ [‘±’].


Q5:   ¿Do the [sub-[axioms-]]systems categories of the arithmetic of the “Wholes & of the arithmetic of the ‘minuses, of the apparently/superficially undirectional, direction-less, or mono-directional counts” as Standard Numbers versus of explicitly other-directional counts’ as Standard Numbers, simply form two irreconcilable, absolutely disparate and separated and, in some ways, diametrically opposite categories?  ¿Or, do we find, in our experience of the human number meme, that these two categories interrelate, and even combine?  ¿That is, does the other-directional counting’ category, m#, the counter-example to, and ‘demotor’ of, the W# concept/definition of Standard Number, which signals the failure of the W# category to constitute the totality of Standard Arithmetic, unite with the category that it counters, demotes, and criticizes, to form an integral new system-of-arithmetic category¿Can our sought-after total explanation/systematization/classification of the ‘meta-system’ of the “Standard Arithmetics” be adequate as either the analysis of the category W# by itself, or the analysis of category m# by itself, seen as radically disjunct alternatives?

A5:  “Yes”, the “Wholes and the ‘minuses are two different, separate, disparate, even qualitatively opposite categories of kinds of number, of number ‘ideo-ontology’. But “No”, they are not absolutely separate, nor absolutely separable, in our experience and knowledge of humanity’s numbers meme. And our theory/presentation of the “Standard Arithmetics” ‘meta-system’ is inadequate to that experience if it takes W# and m# only as dirempt. Indeed, we have encountered an axiomatic system-of-arithmetic category in which, and by which, the three -- the explicitly-[minus-]signed “minus numbers”, the explicitly-[plus-]signed “plus-numbers”, and, at their meeting point, between them, the explicitly-[double-]signed ‘neutral number’ [±0] -- are all integrated into a single system of arithmetic, the system of the integers, Z#. Per this system-category, Z#, we have

Z
  = { ..., -3, -2, -1, ±0, +1, +2, +3, ... }.

The explicit three-fold bi-directional classification of numbers is effected in Z#. That is, each number in Z has either a leftward [-] direction, or a rightward [+] direction, or no direction at all/neither direction, or both directions at once, with equal “weight” [±]. If a pair of ‘oppositely-directioned’ Z#-numbers that are also of equal magnitude [of equal “absolute value”] combine additively, they ‘‘‘mutually annihilate’’’ undoing both their directions and their magnitudes, yielding/leaving/arriving at ±0:

w
+ x2 =
±0, if x2 = -w,

and although it is true that -w is NOT in W, it is also true that -w IS in m, which, in turn, is a subset of Z.

Thus, in this m#-and-W#-synthesizing axioms-system, the equation

w
+ x2 =
±0 can be satisfied.

The composite axioms-system category Z# ‘‘‘contains’’’ seven axioms sub-systems sub-categories: (a.) the 3 past-stages categories N#,a#,& #qaN, which, added together, = W#, which are/is ‘evolutely’ conserved in Z#, plus (b.) m#, the core of Z#, plus; (c.) #qmN, connoting the conversion of the N# into a part of the Z#, plus; (d.) #qma, connoting conversion of a# into another part of Z#, plus; (e.) #qmaN, connoting ‘m#-ization’ of #qaN. We have thusly found our way to a ‘‘‘meta-system transition’’’ [cf. Turchin],

W
#
---) Z#,

a transition within the ‘‘‘meta-system’’’ which is formed by the totality of these Gödelian-Dialectical transitions --

N# ---) W# ---) Z# ---) Q# ---) R# ---) C# ---) H# ---) . . .

-- just as, earlier in this dialogue, we found our way similarly to the ‘‘‘meta-system transition’’’  

N# ---) W#.


Q6:   ¿Does the analysis of this new, composite, successor system-of-Standard-Arithmetic category, Z#, exhaustively systematize our experience and knowledge of the totality of contemporary/modern Standard Arithmetic(s)?  ¿Does that analysis account entirely for, i.e., does it explain, that experience and knowledge? ¿Or are there other categories, already encountered by us unsystematically, in our “chaotic” [Marx] experience and fragmentary knowledge of modern/contemporary Standard Arithmetics, that are required to more fully cover, and to more fully systematize, and to render more fully intelligible, that experience and knowledge; required to classify, and to explain, all of it -- or, at least, more of it: even [at least some of] the most exotic ‘ideo-phenomena’ of modern/contemporary Standard Arithmetic(s) that we have ever encountered?  ¿Are there other categories whose content is not explicitly elaborated by the analysis of the Integers arithmetic-system category, Z#, and of its sub-categories, in the context of the Integers system-category as a whole?  ¿Are there other categories whose analysis/‘content-explicitization’/elaboration would therefore add to/improve/increase the coverage of our theory’s comprehension of [another part of] the totality of the ‘ideo-phenomena’ of modern/contemporary Standard Arithmetic(s)? . . . . . . . . . . . . . . . "



Regards,

Miguel

Saturday, September 22, 2012

A Marxian Approach to the Meaning of Dialectical Equations

Full Title: '''Person-ification''' and 'Im-Person-ation' -- A Marxian Approach to the Meaning of Dialectical-Mathematical Formulae.


Dear Readers,

I thought you might enjoy the following excerpt from Part II. of F.E.D. Vignette #4, "The Goedelian Dialectic of the Standard Arithmetics."

[p. II-10] --

"
‘‘‘Person-ification’’’ and ‘Im-Person-ation’. The physical terms, the tangible ideographical symbols, that the core ‘meta-model’ of this essay sums, are not, in any sense, in themselves, subjects, or agents -- centers that initiate action -- in their own right. How could they be? They are but, e.g., small, solidified pools of toner, adhering to paper.

Of course, to those who share in the ‘‘‘inter-subjectivity’’’ for which these desiccated droplets make meaningful marks, those marks evoke, whenever those ‘sharers’ read them, specific meanings, particular ideas. But these ideas live, so far as we know, only inside individual human minds.

Ideas are vivified only by living human beings, forming them, holding them in mind.

Ideas may have some minimal subconscious, unintentional ‘subject-hood’, some agency, in a human mind, once willfully formed in that mind by action of its ‘mind-er’. But almost all of any ‘agent-hood’, or ‘subject-ness’, that ideas possess, is consciously lent to them by each human subject who forms them in mind, in response to, e.g., human speech, or to some textual symbol(s).

Their ‘subject-ivity’ is borrowed from the real subjects.

Their ‘agent-ness’ persists only when, and only while, they are being ‘‘‘person-ified’’’, or ‘[im-]person-ated’ by a real person.

To believe otherwise is fetishism, that signal symptom of ideology, of the failure of science -- like the “Fetishism of Commodities”, the fetishism of Money, the fetishism of Capital, the fetishism of [exchange-]Value in general -- that Marx so devastatingly diagnosed in the ideology-compromised science of classical capitalist political economy.

To believe otherwise would be a ‘fetishism of Ideas’, akin to the ideology to which Plato’s Socrates -- and to which at least the early Plato as well, prior to The Parmenides -- succumbed: not to mention so many others since!

These physical symbols -- these ‘empapered’ patterns of ink, staining the page -- are dead; a deceased residue of past, ended thought-life, that once guided the hand that wrote down their ‘conventioned’ representatives as marks on ...papyrus..., parchment..., paper, as human mind-remains, ‘psychoartefacts’.

And dead they remain -- unless a living person enlivens them, by comprehendingly reading them, and by [re-]cognizing them”: ‘‘‘impersonating’’’ them -- infusing them with living personality, with living human subjectivity, with active agency -- by thinking them, and therefore also by ‘‘‘incarnating’’’ them in that person’s seemingly flesh-less mind; by ‘‘‘mentally embodying’’’ them, in that person’s seemingly ‘body-less’, ‘dis-embodied’’’ mind -- as acting, interacting, [self-]critiquing and [self-]changing human thoughts, residing, for a time, within the space of self-aware consciousness of a breathing being.

The conclusion with which we are left is that what these symbols really represent are human acts, human cognitive acts -- “Mental Operations” [cf. Boole]. It is people -- human persons -- who animate the Cs and the Ms of Marx’s C<--->M<--->C's and M<--->C<--->M's, who stand behind, and act behind, who ‘enmask’ themselves with -- who “personify” [Marx] -- these “social relations of production”.

Likewise, it is people who animate that which the arithmetical-system category-operators -- N#, a#, m#, f#, etc., denote.

Like each fictional character of a famous novel, made into a movie, the ‘idea-eventities’ which these ‘connotograms’ and ‘categorograms’ conjure in the consciousnesses of their ‘cognizors’ can live and act only if impersonated by a human person, by a human subject, by a human actor.

Objects, including even pre-human/extra-human living ‘[ev]entities’, other biological beings, do not ‘self-awarely’ enact dialectical critique.

To our knowledge, only humans can enact true critique.

Therefore, the dialectical-ideographical symbols employed in this essay, the operator symbols for immanent critique that constitute our main ‘meta-model’, must denote intuitive operations, operations that can only be carried out by human subjects: they denote the operations of human minds.

What these symbols symbolize are [trans-Boolean] human mental movements, ‘‘‘dialogical’’’ andself-dialogical mental activities of human beings.

In the last analysis, the formulae of our ‘meta-model’ evoke a description of, or a guide to, one’s own thought process, one’s own self-dialogue, in process of considering the meaning/definition of number in modern/- contemporary Standard Arithmetic(s).

These formulae are ‘mind-guides’, ‘replayable’ condensed recordings of, e.g., past, polished, proven-to-be-advantageous ‘thought-trains’/ ‘thought-sequences’ / thought-progressions -- ‘‘‘programs’’’/ ‘‘‘software’’’, not for a digital computer, but for a human mind; ‘thought-recipes’ & ‘thought-guides’; scores for symphonies of thought.

Our written-out recordings are a means for presently following the past thoughts of others, or of ourselves, thoughts that left behind a ‘‘‘fossil record’’’ in tangible, written form, a form that can be deciphered/‘re-minded’ to ‘re-navigate’ present readers’ thoughts, anew, down mind-roads of old, on trails blazed before, by others.

If you are “following” the categorial progression/systems-progression modeled herein -- conjuring up for yourself, in your own mind, ‘similants’ of the connotations and intuitions of the axioms-systems that its terms interpret -- then its symbols, its formulae, its equations, are, thereby, now about you.

¡
These symbols are now describing and guiding what is going on in your own mind while you read them, and while you think them!

They are now describing, as well as steering, your own thoughts.

The symbols of this progression of symbols are symbolizing the progression of your own thoughts now. All of this algebra is describing your own mental operations now. All of this ideography is ‘‘‘graphing’’’ the flow of what have become your ideas now.

The formulae that follow -- the human minds behind them -- call out to you to embody them in your own thoughts, to lend them your mind, and to let them orchestrate the flow of your consciousness, just for the time that your beholding of their presentation takes you.

These formulae call you to become them, to “simulate” them in your own inner seeing, to ‘‘‘personify’’’ them, to ‘im-person-ate’ and to ‘im-person-ize’ their intensions and connotations, their meanings, until they have made themselves known to and in you, via the systematic journey of comprehension of modern, standard arithmetic and number along which they are now ready to conduct you."



Regards,


Miguel



Friday, September 21, 2012

The Goedelian Dialectic and Knowledge Representation Condensation

Full Title: The Goedelian Dialectic, Knowledge Representation Condensation, and The Dialectical Operations Stage of Human Adult Cognitive Development.


Dear Readers,

Part II. of II. of F.E.D. Vignette #4, "The Goedelian Dialectic of the Standard Arithmetics", was just posted to the www.dialectics.org website.

The scope and content of Part II. is rather replete, but, for the purposes of this post, I want to quote its discussion of just one topic, that shows how accession to the dialectical operations stage of human adult cognitive development -- transcending the last Piagettian "formal operations" stage -- can open, among many other new vistas, new vistas of knowledge representation condensation.

Below I have reproduced Part II., pages II-58 through II-59, as best as the available typography here will allow.

The definitions of the symbols for the axioms-systems of arithmetic used in the extract below are as follows:

N# connotes the axioms-system of the arithmetic of the "Natural Numbers", N;

W# connotes the axioms-system of the arithmetic of the "Whole Numbers", W;

Z# connotes the axioms-system of the arithmetic of the "Integers" [in German, <<Zahlen>>], Z;

Q# connotes the axioms-system of the arithmetic of the "Rational, or Quotient, Numbers", Q;

R# connotes the axioms-system of the arithmetic of the "Real Numbers", R;

C# connotes the axioms-system of the arithmetic of the "Complex Numbers", C;

H# connotes the axioms-system of the arithmetic of the Hamilton "Quaternions", H;

O# connotes the axioms-system of the arithmetic of the Cayley/Graves "Octonions", O;

K# connotes the axioms-system of the arithmetic of the "William Kingdon Clifford Numbers", K;

G# connotes the axioms-system of the arithmetic of the Grassmann "Geometric Numbers", G, and;

X# connotes the axioms-system of the arithmetic of the [unknown] "next" arithmetic, X, . . .


"Symbolic Economy, Semantic Density / Semantic Productivity, and Mnemonic Power.

The ‘Dialectical Equation’ that constitutes our ‘meta-model’ of the systems of the ‘‘‘Standard Arithmetics’’’--

#
)-|-(s.....=.....(.#N.)^(2^s)


-- functions also as our Encyclopedia Dialectica definition of ‘‘‘Standard Arithmetic’’’.


As such, it is a dialectically ‘‘‘open-ended’’’ kind of definition.


No final term, no ultimate ‘meta-meristem’, no closing ‘‘‘culminant’’’, is specified in this ‘meta-model’, by its ‘Dialectical Equation’.



It remains a ‘“potentially infinite”’ [cf. Aristotle] sequence of series, though one which is always, at any given moment of Terran human history, actually [meta-]finite in terms of that part of its infinite potential which has been actualized by Terran humanity so-far.

This dialectical definition of ‘‘‘Standard Arithmetic’’’ is therefore not simply N#, or W#, or even C#, or H#, or O#.


It is, on the contrary, the entire ‘meta-system-atic’, dialectical and cognitive movement from N# to W#, from W# to Z#, from Z# to Q#, from Q# to R#, from R# to C#, from C# to H#, from H# to O#, and beyond, that is summarized by, and “contained in”, that ‘Dialectical Equation’ --

....N#...---)...W#...---)....Z#...---)....Q#...---)...R#...---)...C#...---)...H#...---)...O#...---)...K#...---)...G#...
s:.1......---).....2.......---).....3......---)......4......---)....5......---).....6......---)....7......---)....8......---).....9......---)..10 ...

-- both actually [e.g., to K# and to G#.], and potentially [to arithmetics beyond those that have been actualized -- codified or axiomatized -- by Terran humanity to-date], given the incompletability or inexhaustibility of mathematics in general, and of arithmetics in particular, established by Gödel.

However, confronted with the potential infinity of such encyclopedic dialectical-equational definitions, we must grapple with issues of the ease and compactness of their representability via our 'dialectical arithmetics, and via their dialectical algebras.


Doing so, we find that our situation is, indeed, quite favorable in that regard.


If we strip the ‘Dialectical Equation meta-model’ that forms the core of this essay down to its bare essentials, stripping off all of the helpful but inessential taxonomical locator epithets, or ‘dialectical diacritical marks’, then our most condensed concentration of the meaning of this entire essay requires just four symbols, or symbolic elements, namely, the elements ‘_’, ‘N’, ‘2’, and ‘6’, arrayed as follows --

N
^(2^6)

-- such that the 4 symbolic-elements above [given font logistics supporting superscripts, and superscripts of superscripts], so arranged, can replace, e.g., the entire 64-term, ~641 symbolic-element expression that concludes the core section of this essay. They can do so in this sense: the entire 64-term series can be re-constituted and recovered, from the 4 symbol, ‘semantically concentrated’ version, simply by repeatedly applying 3 simple rules -- i.e., just 3 of the 9 core axioms of the NQ space of dialectical arithmetic, as given herein within section B.i. -- namely, Axioms §7, §8, and §9 --


§7. For all n in N:...qn + qn....=....qn.

§8. For all j, and k, both in N:...If j is quantitatively unequal to k, then qj + qk is qualitatively unequal to qx for any x in N.
§9. For every j and k, both in N:...qk x qj....=....qj + qk+j.

-- and by one or more applications of the ‘Organonic Algebraic Method’ to ‘‘‘re-solve-for’’’ any once-known but no-longer-known /-remembered terms, when the meanings of some of them are forgotten subsequent to reading this essay.


If we take the ‘‘‘replacement rate’’’ -- the percent-ratio of the count of the number of terms replaced to that of the symbolic elements so replacing -- as a crude metric for the degree of semantic compression, or of knowledge-representation-condensation, achieved, then the ‘semantic density’ improvements that we are achieving by using the stripped down, dialectical, Dyadic Seldon Function formulations, are impressive, viz. --

· out to R# and its «aporia», N^(2^5):...............32/4...=...........8 =............800% semantic condensation rate;
· out to C# and its «aporia», N^(2^6):................64/4...=........16 =.......1,600%semantic condensation rate;
· out to H# and its «aporia», N^(2^7):.............128/4...=........32 =.......3,200%semantic condensation rate;
· out to O# and its «aporia», N^(2^8):.............256/4...=........64 =.......6,400%semantic condensation rate;
· out to K# and its «aporia», N^(2^9):.............512/4...=......128 =...12,800% semantic condensation rate;
· out to G# and its «aporia», N^(2^10):...1,024/5...~.......204...=..20,400%semantic condensation rate;
· out to X# and its «aporia», N^(2^11):...2,048/5...~.......410...=...41,000%semantic condensation rate;

Using the ‘minimalized’ Seldon Function format -- a^(n^s) -- the systematic(s) core of a discourse: of a whole lecture, or of a whole text -- paper, essay, book, multi-«buch»/multi-volume treatise, etc. ... -- can be mnemonically summarized, using as few as four symbolic elements, in an expression which, with the application of three rules, & of the ‘organonic method’, if needed, can, at will, be quickly reconstituted into a series/sum/cumulum of tens, or hundreds, or thousands,... of terms, capturing, in systematically-ordered detail, the gist of the content of that discourse.

That ‘minimalized’ Seldon Function format can formulate condensed, ‘re-implicitized’, ‘connotationally curtailed’, or ‘darkened’, ‘‘‘black [w]holes’’’ of information, from which ‘‘‘white [w]holes’’’ of outpouring ‘‘‘[w]holistic’’’/mnemonic re-elaboration and reconstitution of that information are ever ready to be ‘re-unfolded’, to be ‘re-unfurled’, to be ‘rotely’ ‘re-burgeoned’, by those who know the 3 axiomatic rules [and the ‘organonic method’].

The mere assertion of a category, within a specific, interpreted progression/sum, or ‘[ac]cumulum’, of categories, is not, in itself, the delineation and articulation, or ‘explicitization’, of the detailed content -- of the progression/sum/cumulum of sub-categories and of sub-sub-categories... which are implicit in that category when it is asserted as an unarticluated, undelineated, undivided, univocal whole.

But the assertion of that undivided category does serve as a collective name for, and as a reminder of -- an intimation of -- the content of that category in its more fully articulated detail, as experienced/conducted in the past, and as still ‘rememberable’, to some degree, by the user, presently.


Of course, in the last analysis, the ‘categorogram’ or ‘category ideogram’ symbols, that constitute these dialectical progression expressions, are intensional symbols”, not extensional symbols”. Each is a ‘connotogram’, not an explicit list of symbols in 1-to-1 correspondence with “every last” element of meaning of the [ideo-]ontological category that it represents.

The meanings of those ‘categorograms’ are not “all there in the symbols”, and such they never can be. What each is, is a ‘mnemonic trigger’, an ‘associational catalyst’, to remind the user of, and to help [re-]evoke in the user, the rich totality of ‘implicit semanticities’ that these “intensional” symbols intend.

The richer the web of associations, of previously constructed and ‘re-member-éd’ knowledge -- of remembered experience in general -- that the user brings to those symbols, evoked in the user’s past, and retained in mind, i.e., in the user’s ‘meme-ory’, ever since, the richer, then, the totality of meanings that these ‘semantically densified’ and ‘semantically concentrated’ symbols ‘‘‘hold’’’ for that user, and the better the odds for that user to evoke that richness in and for others."



Enjoy Part II.!

Regards,

Miguel